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Parikh vectors
8VLT_1 2BIG_1 3MGY_1 Letter Amino acid
6 29 14 N Asparagine
7 13 27 D Aspartic acid
8 23 15 Q Glutamine
6 23 22 I Isoleucine
33 25 20 S Serine
13 29 25 A Alanine
4 10 19 R Arginine
4 11 11 M Methionine
14 22 19 K Lycine
11 14 15 Y Tyrosine
7 24 23 E Glutamic acid
19 22 15 G Glycine
16 39 43 L Leucine
6 15 13 F Phenylalanine
15 14 17 P Proline
19 18 19 T Threonine
6 2 5 W Tryptophan
20 18 22 V Valine
4 2 4 C Cysteine
4 8 12 H Histidine 

8VLT_1|Chain A[auth H]|2526 Fab Heavy Chain|Homo sapiens (9606)
>2BIG_1|Chains A, B|PHOSPHOSERINE AMINOTRANSFERASE|BACILLUS ALCALOPHILUS (1445)
>3MGY_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
8VLT , Knot 98 222 0.79 40 152 209 
EVQLVQSGAEVKKPGESLKISCKTSGYSFTNHWIAWVRQMPGAGLEWMGMIYPHDSDTRYNPSFQGLVTISVDKSITTAYLQWRSLKASDTAMYYCARPPFSGSYEGLDYWGQGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPK
2BIG , Knot 154 361 0.83 40 202 341 
MVKQVFNFNAGPSALPKPALERAQKELLNFNDTQMSVMELSHRSQSYEEVHEQAQNLLRELLQIPNDYQILFLQGGASLQFTMLPMNLLTKGTIGNYVLTGSWSEKALKEAKLLGETHIAASTKANSYQSIPDFSEFQLNENDAYLHITSNNTIYGTQYQNFPEINHAPLIADMSSDILSRPLKVNQFGMIYAGAQKNLGPSGVTVVIVKKDLLNTKVEQVPTMLQYATHIKSDSLYNTPPTFSIYMLRNVLDWIKDLGGAEAIAKQNEEKAKIIYDTIDESNGFYVGHAEKGSRSLMNVTFNLRNEELNQQFLAKAKEQGFVGLNGHRSVGGCRASIYNAVPIDACIALRELMIQFKENA
3MGY , Knot 159 360 0.86 40 224 346 
MSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(8VLT_1)}(2) \setminus P_{f(2BIG_1)}(2)|=58\), \(|P_{f(2BIG_1)}(2) \setminus P_{f(8VLT_1)}(2)|=108\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:010110011010011001010000010010001111100111111011111010000000010101110101000100101010010100011000101110100011001101011010010001101111110000001101111011000110110101001110011001111100011001001101100011000010010001000010001010
Pair \(Z_2\) Length of longest common subsequence
8VLT_1,2BIG_1 166 4
8VLT_1,3MGY_1 190 4
2BIG_1,3MGY_1 152 4

Newick tree

 
[
	8VLT_1:93.18,
	[
		2BIG_1:76,3MGY_1:76
	]:17.18
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{583 }{\log_{20} 583}-\frac{222}{\log_{20}222})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 8VLT_1 2BIG_1 127 102
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]